In this post, we will see the book Elements Of Applied Mathematics by Ya. B. Zeldovich and A. D. Myskis.

This book is not a textbook in the ordinary sense of the word but rather a reader in the mathematical sciences. Using simple examples taken from physics and a variety of mathematical problems, we have tried to introduce the reader to a broad range of ideas and methods that are found in present-day applications of mathematics to physics, engineering and other fields. Some of these ideas and methods (such as the use of the delta function, the principle of superposition, obtaining asymptotic expressions, etc.) have not been sufficiently discussed in the ordinary run of mathematics textbooks for non-mathematicians, and so this text can serve as a supplement to such textbooks. Our aim has been to elucidate the basic ideas of mathematical methods and the general laws of the phenomena at hand. Formal proofs, exceptions and complicating factors have for the most part been dropped. Instead we have strived in certain places to go deeper into the physical picture of the processes.

The book was first published by Mir in 1976 and was translated from the Russian by George Yankovsky.

PDF | 666 pages | Cover | OCR

All credits to the original uploader Siddharth for this book.

The Internet Archive link for the book.

Contents

PREFACE 9

Chapter 1. Certain Numerical Methods 13

1.1 Numerical integration 14
1.2 Computing sums by means of integrals 20
1.3 Numerical solution of equations 28 Answers and Solutions 36

Chapter 2. Mathematical Treatment of Experimental Results 39

2.1 Tables and differences 39
2.2 Integration and differentiation of tabulated functions 44
2.3 Fitting of experimental data by the least-squares method 49
2.4 The graphical method of curve fitting 55
Answers and Solutions 62

Chapter 3. More on Integrals and Series 65

3.1 Improper integrals 65
3.2 Integrating rapidly varying functions 73
3.3 Stirling’s formula 82
3.4 Integrating rapidly oscillating functions 84
3.5 Numerical series 88
3.6 Integrals depending on a parameter 99
Answers and Solutions 103

Chapter 4. Functions of Several Variables 108

4.1 Partial derivatives 108
4.2 Geometrical meaning of a function of two variables 115
4.3 Implicit functions 118
4.4 Electron tube 126
4.5 Envelope of a family of curves 129
4.6 Taylor’s series and extremum problems 131
4.7 Multiple integrals 139
4.8 Multidimensional space and number of degrees of freedom 150
Answers and Solutions 154

Chapter 5. Functions of a Complex Variable 158

5.1 Basic properties of complex numbers 158
5.2 Conjugate complex numbers 161
5.3 Raising a number to an imaginary power. Euler’s formula 164
5.4 Logarithms and roots 169
5.5 Describing harmonic oscillations by the exponential function of an imaginary argument 173
5.6 The derivative of a function of a complex variable 180
5.7 Harmonic functions 182
5.8 The integral of a function of a complex variable 184
5.9 Residues 190
Answers and Solutions 199

Chapter 6. Dirac’s Delta Function 203

6.1 Dirac’s delta function a(x) 203
6.2 Green’s function 208
6.3 Functions related to the delta function 214
6.4 On the Stieltjes integral 221
Answers and Solutions 223

Chapter 7. Differential Equations 225

7.1 Geometric meaning of a first-order differential equation 225
7.2 Integrable types of first-order equations 229
7.3 Second-order homogeneous linear equations with constant coefficients 236
7.4 A simple second-order nonhomogeneous linear equation 242
7.5 Second-order nonhomogeneous linear equations with constant coefficients 249
7.6 Stable and unstable solutions 256
Answers and Solutions 261

Chapter 8. Differential Equations Continued 263

8.1 Singular points 263
8.2 Systems of differential equations 265
8.3 Determinants and the solution of linear systems with constant coefficients 270
8.4 Lyapunov stability of the equilibrium state 274
8.5 Constructing approximate formulas for a solution 277
8.6 Adiabatic variation of a solution 285
8.7 Numerical solution of differential equations 288
8.8 Boundary-value problems 297
8.9 Boundary layer 303
8.10 Similarity of phenomena 305
Answers and Solutions 309

Chapter 9. Vectors 312

9.1 Linear operations on vectors 313
9.2 The scalar product of vectors 319
9.3 The derivative of a vector 321
9.4 The motion of a material point 324
9.5 Basic facts about tensors 328
9.6 Multidimensional vector space 333
Answers and Solutions 336

Chapter 10. Field Theory 340

10.1 Introduction 340
10.2 Scalar field and gradient 341
10.3 Potential energy and force 345
10.4 Velocity field and flux 351
10.5 Electrostatic field, its potential and flux 356
10.6 Examples 359
10.7 General vector field and its divergence 369
10.8 The divergence of a velocity field and the continuity equation 374
10.9 The divergence of an electric field and the Poisson equation 376
10.10 An area vector and pressure 379
Answers and Solutions 384

Chapter 11. Vector Product and Rotation 388

11.1 The vector product of two vectors 388
11.2 Some applications to mechanics 392
11.3 Motion in a central-force field 396
11.4 Rotation of a rigid body 406
11.5 Symmetric and antisymmetric tensors 408
11.6 True vectors and pseudovectors 415
11.7 The curl of a vector field 416
11.8 The Hamiltonian operator del 423
11.9 Potential fields 426
11.10 The curl of a velocity field 430
11.11 Magnetic field and electric current 433
11.12 Electromagnetic field and Maxwell’s equations 438
11.13 Potential in a multiply connected region 442
Answers and Solutions 445

Chapter 12. Calculus of Variations 450

12.1 An instance of passing from a finite number of degrees of freedom to an infinite number 450
12.2 Functional 456
12.3 Necessary condition of an extremum 460
12.4 Euler’s equation 462
12.5 Does a solution always exist? 468
12.6 Variants of the basic problem 474
12.7 Conditional extremum for a finite number of degrees of freedom 476
12.8 Conditional extremum in the calculus of variations 479
12.9 Extremum problems with restrictions 488
12.10 Variational principles. Fermat’s principle in optics 491
12.11 Principle of least action 499
12.12 Direct methods 503
Answers and Solutions 508

Chapter 13. Theory of Probability 514

13.1 Statement of the problem 514
13.2 Multiplication of probabilities 517
13.3 Analysing the results of many trials 522
13.4 Entropy 533
13.5 Radioactive decay. Poisson’s formula 539
13.6 An alternative derivation of the Poisson distribution 542
13.7 Continuously distributed quantities 544
13.8 The case of a very large number of trials 549
13.9 Correlational dependence 556
13.10 On the distribution of primes 561
Answers and Solutions 567

Chapter 14. Fourier Transformation 573

14.1 Introduction 573
14.2 Formulas of the Fourier transformation 577
14.3 Causality and dispersion relations 585
14.4 Properties of the Fourier transformation 589
14.5 Bell-shaped transformation and the uncertainty principle 597
14.6 Harmonic analysis of a periodic function 602
14.7 Hilbert space 606
14.8 Modulus and phase of spectral density 612
Answer and Solutions 615

Chapter 15. Digital Computers 619

15.1 Analogue computers 619
15.2 Digital computers 621
15.3 Representation of numbers and instructions in digital computers 623
15.4 Programming 628
15.5 Use computers1 634
Answers and Solutions 642
REFERENCES 645
INDEX 696